Upscaling and spatial localization of non-local energies with applications to crystal plasticity

We describe multiscale geometrical changes via structured deformations [Formula: see text] and the non-local energetic response at a point x via a function [Formula: see text] of the weighted averages of the jumps [Formula: see text] of microlevel deformations [Formula: see text] at points y within a distance r of x. The deformations [Formula: see text] are chosen so that [Formula: see text] and [Formula: see text]. We provide conditions on [Formula: see text] under which the upscaling “[Formula: see text]” results in a macroscale energy that depends through [Formula: see text] on (1) the jumps [Formula: see text] of g and the “disarrangement field”[Formula: see text], (2) the “horizon” r, and (3) the weighting function [Formula: see text] for microlevel averaging of [Formula: see text]. We also study the upscaling “[Formula: see text]” followed by spatial localization “[Formula: see text]” and show that this succession of processes results in a purely local macroscale energy [Formula: see text] that depends through [Formula: see text] upon the jumps [Formula: see text] of g and the “disarrangement field”[Formula: see text] alone. In special settings, such macroscale energies [Formula: see text] have been shown to support the phenomena of yielding and hysteresis, and our results provide a broader setting for studying such yielding and hysteresis. As an illustration, we apply our results in the context of the plasticity of single crystals.